Elsevier

Discrete Mathematics

Volume 341, Issue 5, May 2018, Pages 1253-1263
Discrete Mathematics

Stability in the Erdős–Gallai Theorem on cycles and paths, II

https://doi.org/10.1016/j.disc.2017.12.018Get rights and content
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Abstract

The Erdős–Gallai Theorem states that for k3, any n-vertex graph with no cycle of length at least k has at most 12(k1)(n1) edges. A stronger version of the Erdős–Gallai Theorem was given by Kopylov: If G is a 2-connected n-vertex graph with no cycle of length at least k, then e(G)max{h(n,k,2),h(n,k,k12)}, where h(n,k,a)ka2+a(nk+a). Furthermore, Kopylov presented the two possible extremal graphs, one with h(n,k,2) edges and one with h(n,k,k12) edges.

In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for k3 odd and all nk, every n-vertex 2-connected graph G with no cycle of length at least k is a subgraph of one of the two extremal graphs or e(G)max{h(n,k,3),h(n,k,k32)}. The upper bound for e(G) here is tight.

Keywords

Turán problem
Cycles
Paths

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This paper started at SQuaRES meeting of the American Institute of Mathematics.